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Find the most general antiderivative of the function (Check our Rsa by ditterentiatlon. Use for the constant 0f tne uaidarivative.5)e* = 6)F6)...

Question

Find the most general antiderivative of the function (Check our Rsa by ditterentiatlon. Use for the constant 0f tne uaidarivative.5)e* = 6)F6)

Find the most general antiderivative of the function (Check our Rsa by ditterentiatlon. Use for the constant 0f tne uaidarivative. 5)e* = 6) F6)



Answers

Find the most general antiderivative of the function
(Check your answer by differentiation
$$f(x)=5 e^{x}-3 \cosh x$$

We were given the following function. When you're asked to find anti dresses off this given function and you were asked a checker work, we're using differentiation. So to start off less, we can recognize that we have tea on the top and sheer no bottom so we can take is to this team in the fourth and bring it back up. So we are left with three minus one over tea plus six over T squirt, which we know is equal to three miners to you to the negative one plus 62 the negative chip. So in order to figure out what our anti derivative is, we can use our reverse power roll. So we have property is equal to a man we know not the anti derivative constantly. Just add in the variable, um, minus one over cheese is just the natural log of absolute valley of tea. And then we have 63 night of when we add one to that. Well, give us to tonight. They want anyone divide six by neighborhood, One which will get us now six over tea. And of course, you cannot forget our constant for anti derivatives. So we get a posse when I do a bathroom and checker work to make sure we have done it correctly by differentiating to try to get back to your original question. So we have the anti er the derivative of three teens, just three derivative of natural. Of that's without is he is just one over tea. And then when we friend and the derivative of six over t, we just have six TT negative one. So you must buy maybe one by negative six sort against positive six t to the negative too, so we can see that this matches up with our original function. So therefore, this is our anti derivative.

Were given a falling function. Ever have access he could accept that might accept their plus two X All were extra. 14 were asked on inter derivative of this function. We were asked to check her work using differentiation. So shut off with Let's bring X 1/4 up genuine writer. So I looked with X minus exit native one plus two x 2 93 And so then we can use a reverse power role to find our derivatives. So we have our trainer Inter derivative. So we have 1/2 x squared and then we know that the interpretive off accident of one inches national log of the absolute value of X Then we have to accidentally three. So they bump up trucks and entry to trucks amended to and then we divide to buy chew By night of choose, we get minus X who didn't go to? If we can then right? I was X squared over two Mr Natural log of severely vax There's one over x squared. We're also can not forget our constant whoever with on anti terror of this so that we can differentiate this entire thing we get X my ass vaccinated one plus two x in 93 plus zero when you see that this much is up with original function. So therefore we know that this is anti drifted.

We want to find the most general anti derivative for this here. So the first thing we can do is use the fact that the integral is a linear operator, which means we can distribute this across plus and minus signs. So first, we'd write this as the interval of five X minus interval of six X and the other property of anti derivatives is we can factor out Constance so we can rewrite this as minus six. Love integral up X now, integrating a constant, we just multiply by our variable. So in this case, X and then to integrate X Well, this is really extra the first power. So this just follows from power rule. So there should be X to the one plus our new power plus one. So we get the new power of two and then we divide by our new power and then they tell us we need to add a arbitrary constant See to this. So let's go ahead and simplify this down a little. So the two of the six simplified to three. So that should give us five x minus three x squared plus C. And so this here would be the most general anti derivative

This question asked us to find the anti derivative Before we take the anti derivative. I would highly recommend writing this with your fractions to the 1/2 instead of square root and makes a lot easier to read. So, as you can see instead of six, squared of ex would have six after the 1/2 because extra the 1/2 of the same thing a squirt of X and then likewise for 16 is the same thing is cute of of six through of acts not to find the anti derivative what we know we're looking for. As we're gonna take X, we're gonna increase the exponents by one which gives us three divided by two and then 3/2 times what gives us sex. Answer his fork. Increase the expert for 1/6 by 116 plus 6 67 sex and then we have 67 a czar coefficient plus c. We take the derivative the check. You get 3/2 times four, which is six x Subtract Expert, Buy one, get 1/2 minus. This cancels just get X and I subtract expert by one. You got one


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